The unirationality of Hurwitz spaces of 6-gonal curves of small genus

TL;DR

This paper proves the unirationality of Hurwitz spaces for 6-gonal curves with small genus using liaison constructions and finite field computations, extending known results to a range of genera.

Contribution

It establishes the unirationality of Hurwitz spaces for 6-gonal curves of genus 5 to 28 and some higher genera, employing a novel liaison construction approach.

Findings

Unirationality proven for specified genus ranges

Liaison construction in b6^1 imes \u0000b6^2 used

Finite field computations support the proof

Abstract

In this short note we prove the unirationality of Hurwitz spaces of 6-gonal curves of genus $g$ with $5\leq g\leq 28$ or $g=30,31,33,35,36,40,45$. Key ingredient is a liaison construction in $\PP^1 \times \PP^2$. By semicontinuity, the proof of the dominance of this construction is reduced to a computation of a single curve over a finite field.